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I'm about to install Mint on a box with an Intel CoreTM2 Duo E8400, 4Gb RAM and an Nvidia 9400GT.

Looking at the editions available, I am not sure whether to go for the standard or x64 versions. The site says that the x64 version is optimised for 64bit, but that the standard edition is also compatible and is more stable.

So which one should I use and what are the differences in performance?

Nowadays the 64 bit version of Mint is really no less stable then the 32 bit version. You just have to make sure that the apps you use are compiled for 64 bit machines. You can also run the 32 bit version of apps. if you have the correct libraries installed. Once you see the performance increase you'll be very happy.